Terminology

The shortest distance on the surface of a solid is generally termed a geodesic, be it an ellipsoid of revolution, aposphere, etc. On a sphere, the geodesic is termed a Great Circle.

HOWEVER, when computing the distance between two points using a projected coordinate system, that is a conformal projection such as Transverse Mercator, Oblique Mercator, Normal Mercator, Stereographic, or Lambert Conformal Conic - that then is a GRID distance which can be converted to an equivalent GEODETIC distance using the function for "Scale Factor at a Point." The conversion is then termed "Grid Distance to Geodetic Distance," even though it will not be as exactly correct as a true ellipsoidal geodesic. Closer to the truth with a TM than with a Lambert or other conformal projection, but still not exactly "on."

So, it can be termed "geodetic distance" or a "geodesic distance," depending on just how you got there ...

The Math

Spherical Approximation

The simplest way to compute geodesics is using a sphere as an approximation for the earth. This from Mikael Rittri on the Proj mailing list:

If 1 percent accuracy is enough, I think you can use spherical formulas with a fixed Earth radius.
You can find good formulas in the Aviation Formulary of Ed Williams, ​http://williams.best.vwh.net/avform.htm.

For the fixed Earth radius, I would choose the average of the:

c = radius of curvature at the poles,
b2 / a = radius of curvature in a meridian plane at the equator,

since these are the extreme values for the local radius of curvature of the earth ellipsoid.

When computing the azimuth between two points by the spherical formulas, I think the
maximal error on WGS84 will be 0.2 degrees, at least if the points are not too far away
(less than 1000 km apart, say). The error should be maximal near the equator, for azimuths near northeast etc.

I am not sure about the spherical errors for the forward geodetic problem:

point positioning given initial point, distance and azimuth.

Ellipsoidal Approximation

For more accuracy, the earth can be approximated with an ellipsoid, complicating the math somewhat. See the wikipedia page, ​Geodesics on an ellipsoid, for more information.

Earlier Mr. Evenden had posted to the PROJ.4 mailing list this code for determination of true distance and respective forward and back azimuths between two points on the ellipsoid. Good for any pair of points that are not antipodal.
Later he posted that this was not in fact the translation of NGS FORTRAN code, but something else. But, for what it's worth, here is the posted code (source unknown):

PROJ.4 - geod program

The PROJ.4 geod program can be used for great circle distances on an ellipsoid. As of proj verion 4.9.0, this uses a translation of GeographicLib::Geodesic (see below) into C. The underlying geodesic calculation API is exposed as part of the PROJ.4 library (via the geodesic.h header). Prior to version 4.9.0, the algorithm documented here was used:

Triaxial Ellipsoid

A triaxial ellipsoid is a marginally better approximation to the shape of the earth
than an ellipsoid of revolution.
The problem of geodesics on a triaxial ellipsoid was solved by Jacobi in 1838.
For a discussion of this problem see